Question

The table shows the amount of pet food in cups remaining in an automatic feeder as a function of the number of meals the feeder has dispensed.

automatic feeder
Number of Meals Dispensed Amount of Pet Food Remaining (cups)
1 21
3 15
6 6
7 3
Based on the table, which function models this situation?
f (x)=

Answer 1

To determine a function that models the relationship between the number of meals dispensed and the amount of pet food remaining, we can start by analyzing the data provided in the table:

\[
\begin{array}{|c|c|}
\hline
\text{Number of Meals Dispensed (x)} & \text{Amount of Pet Food Remaining (f(x)) (cups)} \\
\hline
1 & 21 \\
3 & 15 \\
6 & 6 \\
7 & 3 \\
\hline
\end{array}
\]

Let's look for a pattern in the values. Observe the amount of pet food remaining after each meal:

1. From 21 cups to 15 cups (1 meal to 3 meals): a decrease of 6 cups over 2 meals.
2. From 15 cups to 6 cups (3 meals to 6 meals): a decrease of 9 cups over 3 meals.
3. From 6 cups to 3 cups (6 meals to 7 meals): a decrease of 3 cups over 1 meal.

Now, calculate the total cups used after each meal dispensed:

1. After 1 meal (6 cups used): 21 - 6 = 15
2. After 3 meals (9 cups used): 15 - 9 = 6
3. After 6 meals (3 cups used): 6 - 3 = 3

It appears that the pet food decreases by different amounts, but let's check the total food dispensed more rigorously.

We can create a linear function if the amounts remaining decrease consistently and can be modeled. Checking the values, we note:

- Start with 21 cups.
- After 1 meal: \(21 - 6 = 15\) (6 cups per 2 meals)
- After 3 meals: \(15 - 9 = 6\) (9 cups per 3 meals)
- After 6 meals: \(6 - 3 = 3\) (3 cups per 1 meal)

It seems like there is a piecewise nature but also confers a linear model could be applicable with downward growth.

Given the decreasing amounts, we can fit a linear function \(f(x) = mx + b\) where:
- \(m\) is the slope.
- \(b\) is the y-intercept or the amount at \(f(0)\).

To get a rough estimate of linear fit we can use the points:
- \((1, 21)\)
- \((3, 15)\)

Calculating slope \(m\):
\[
m = \frac{f(3) - f(1)}{3 - 1} = \frac{15 - 21}{2} = -3
\]

Using \(b\) from \(f(1) = m(1) + b\):
\[
21 = -3(1) + b \implies b = 24
\]

Linear function becomes:
\[
f(x) = -3x + 24
\]

We can quickly check if this fits remaining cups at other points:
- For \(x=6\): \(f(6) = -3(6) + 24 = -18 + 24 = 6\) (correct)
- For \(x=7\): \(f(7) = -3(7) + 24 = -21 + 24 = 3\) (correct)

Thus, the function modeling the amount of pet food remaining in the feeder, based on the number of meals dispensed, is:
\[
f(x) = -3x + 24
\]